Geometric Algebra 9. Unification Dr Chris Doran ARM Research

Euclidean geometry Represent the Euclidean point x by null vectors Distance is given by the inner product Read off the Euclidean vector Depends on the concept of the origin

Spherical geometry Suppose instead we form Unit vector in an n+1 dimensional space Instead of plotting points in Euclidean space, we can plot them on a sphere No need to pick out a preferred origin any more

Spherical geometry Spherical distance Same pattern as Euclidean case ‘Straight’ lines are now Invariance group are the set of rotors satisfying Generators satisfy The term now becomes essentially redundant and drops out of calculations Left with standard rotors in a Euclidean space. Just rotate the unit sphere

non-Euclidean geometry Historically arrived at by replacing the parallel postulate ‘Straight’ lines become d-lines. Intersect the unit circle at 90o Model this in our conformal framework Unit circle d-lines d-line between X and Y is Translation along a d-line generated by Rotor generates hyperbolic transformations

non-Euclidean geometry Generator of translation along the d-line. Use this to define distance. Write Unit time-like vectors Boost factor from special relativity Distance in non-Euclidean geometry

non-Euclidean distance Distance expands as you get near to the boundary Circle represents a set of points at infinity This is the Poincare disk view of non-Euclidean geometry

non-Euclidean circles Formula unchanged from the Euclidean case Still have Non-Euclidean circle Definition of the centre is not so obvious. Euclidean centre is Reverse the logic above and define

Unification Conformal GA unifies Euclidean, projective, spherical, and hyperbolic geometries in a single compact framework.

Geometries and Klein Understand geometries in terms of the underlying transformation groups Euclidean Affine Projective Conformal Mobius /Inversive Spherical non-Euclidean

Geometries and Klein Projective viewpoint Conformal viewpoint Euclidean Euclidean Affine Spherical Projective non- Euclidean Conformal Conformal Euclidean Affine

Groups Have seen that we can perform dilations with rotors Every linear transformation is rotation + dilation + rotation via SVD Trick is to double size of space Null basis Define bivector Construct group from constraint Keeps null spaces separate. Within null space give general linear group.

Unification Every matrix group can be realised as a rotor group in some suitable space. There is often more than one way to do this.

Design of mathematics Coordinate geometry Complex analysis Vector calculus Tensor analysis Matrix algebra Lie groups Lie algebras Spinors Gauge theory Grassmann algebra Differential forms Berezin calculus Twistors Quaternions Octonions Pauli operators Dirac theory Gravity…

Spinors and twistors Spin matrices act on 2-component wavefunctions These are spinors Very similar to qubits Roger Penrose has put forward a philosophy that spinors are more fundamental than spacetime Start with 2-spinors and build everything up from there

Twistors Look at dimensionality of objects in twistor space Conformal GA of spacetime!

Forms and exterior calculus Working with just the exterior product, exterior differential and duality recovers the language of forms Motivation is that this is the ‘non-metric’ part of the geometric product Interesting development to track is the subject of discrete exterior calculus This has a discrete exterior product Challenge – can you do better? This is associative! Hard to prove.

Resources geometry.mrao.cam.ac.uk chris.doran@arm.com cjld1@cam.ac.uk @chrisjldoran #geometricalgebra github.com/ga